4.2.1. Homogeneous second order linear differential equations with constant coefficients

The differential equation y "+ p · y '+ q · y = 0 , where p and q are constant, is called a homogeneous second-order linear equation.

The form of the general solution y _{0 of a} homogeneous second-order linear equation depends on the roots k ₁ , k _{2 of the} characteristic equation:

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1. If k ₁ , k ₂ are real numbers, and k ₁ ≠ k ₂ , then the general solution is:
2. If k ₁ , k ₂ are real numbers, and k ₁ = k ₂ , then the general solution will be: .
3. If k ₁ , k ₂ are complex numbers, where , then the general solution is: